I am trying to solve the following problem:
Let $\{f_n\}$ be a sequence of measurable functions defined on a probability space, such that
$$
P\left(f_n=\frac1n\right)=1-P(f_n=0)=\frac1n.
$$
Does $\sum_{n=1}^\infty f_n$ converge outside a set of measure zero?
A similar problem was solved here: Does the sum of sequence of measurable functions converge outside a set of measure zero?. The only difference is that in my problem $P(f_n=1/n)$ and $P(f_n=0)$ are not summable, so the Borel-Cantelli lemma does not apply. Is there another way to solve this?
Best Answer
$Ef_n=\frac 1n P(f_n=\frac 1n)+0P(f_n=0)=\frac 1{n^{2}}$. Hence $\sum Ef_n=\sum \frac 1 {n^{2}} <\infty$. By Tonelli's Theorem this implies $E \sum f_n <\infty$. In turn this implies $\sum f_n <\infty$ almost surely.